Heat 1/4 cup olive oil in a large, high-sided sauté pan over medium heat until shimmering. Add the shallot and cook, stirring occasionally, under softened and translucent, 2 to 3 minutes. Add the zucchini, the remaining 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper. Cook, stirring occasionally, until the zucchini cooks down, caramelizes, and softens considerably, about 15 minutes. Reduce the heat if the zucchini starts to burn. The mixture should be jammy and spreadable. Remove from the heat.